The truncated Fourier Operator. IV

TL;DR

This paper investigates the formal prolate spheroid differential operator on a finite symmetric interval, identifying all self-adjoint boundary conditions and highlighting the unique condition that commutes with the truncated Fourier operator.

Contribution

It characterizes all self-adjoint boundary conditions for the operator and identifies the unique one commuting with the truncated Fourier operator.

Findings

Only one boundary condition yields a self-adjoint operator commuting with the truncated Fourier operator.

Complete classification of self-adjoint boundary conditions for the prolate spheroid operator.

Identification of the unique boundary condition related to the Fourier operator.

Abstract

We consider the formal prolate spheroid differential operator on a finite symmetric interval and describe all its self-adjoint boundary conditions. Only one of these boundary conditions corresponds to a self-operator differential operator which commutes with the Fourier operator truncated on the considered finite symmetric interval.